02/29/2016, 10:31 PM
For simplicity and useful properties , let us say we want
Sums and products to be Well defined.
Sums and products to be commutative and associative.
No zero-divisors.
Algebraicly closed.
Not isomorphic to known numbers.
There are 3 types of thought for such number systems.
Although they are all similar.
1) analytic
This leads to stuff like double periodic meromorphic functions.
So f(a+b i) has a lot of cancellation / copies.
We could even work with polar notation and have 3 invariants.
But in essense we end Up with complex numbers and functions.
Or the reals.
This tends to lead to functions alone.
Notice that higher dimensions tend to zero-divisors or loose the Desired properties.
2) geometric
Things like giving the same value for everything ( points ) on a sphere.
Related to modular , norms , determinants and bicomplex , although those examples are of algebraic or algebraic-geo nature.
3) algebraic
Defining sums and products is a typical way.
Groups and rings. Group rings , isomorphisms.
That kind of stuff.
I got inspired by the riemann surface of the log and the hyperoperators to study my numbers. I consider them of this " 3) algebraic " type.
Z1,Z2 are complex Numbers.
R1,R2 are reals.
Tommy's spiral Numbers
---------------------------
(z1,r1) * (z2,r2) = (z1 z2, r1 + r2).
(z1,r1) + (z2,r2) = (z1 + z2,[r1 + r2]\2).
Notice they satisfy the properties AND the distributive property.
Also the complex are a subset of them ( r1 and r2 = 0 ) , what explains the algebraic closure alot.
Here are some links
http://math.eretrandre.org/tetrationforu...p?tid=1037
http://math.eretrandre.org/tetrationforu...p?tid=1036
In general the number of variables is the dimension.
Reductions can lower the dimension but normally with An integer amount.
For instance the groupring R+(C_4) - iso to the couple (z,r).
Or R+(C_3) iso to the complex Numbers.
This is the closest i could find as An answer to your question for commutative Numbers with Nice properties ...
How this is gonna relate to tetration ...
¯\_(ツ)_/¯
Maybe if you had a plan to relate numbers in Some way ...
Anyway hope this enlightens you.
Regards
Tommy1729
The master
Sums and products to be Well defined.
Sums and products to be commutative and associative.
No zero-divisors.
Algebraicly closed.
Not isomorphic to known numbers.
There are 3 types of thought for such number systems.
Although they are all similar.
1) analytic
This leads to stuff like double periodic meromorphic functions.
So f(a+b i) has a lot of cancellation / copies.
We could even work with polar notation and have 3 invariants.
But in essense we end Up with complex numbers and functions.
Or the reals.
This tends to lead to functions alone.
Notice that higher dimensions tend to zero-divisors or loose the Desired properties.
2) geometric
Things like giving the same value for everything ( points ) on a sphere.
Related to modular , norms , determinants and bicomplex , although those examples are of algebraic or algebraic-geo nature.
3) algebraic
Defining sums and products is a typical way.
Groups and rings. Group rings , isomorphisms.
That kind of stuff.
I got inspired by the riemann surface of the log and the hyperoperators to study my numbers. I consider them of this " 3) algebraic " type.
Z1,Z2 are complex Numbers.
R1,R2 are reals.
Tommy's spiral Numbers
---------------------------
(z1,r1) * (z2,r2) = (z1 z2, r1 + r2).
(z1,r1) + (z2,r2) = (z1 + z2,[r1 + r2]\2).
Notice they satisfy the properties AND the distributive property.
Also the complex are a subset of them ( r1 and r2 = 0 ) , what explains the algebraic closure alot.
Here are some links
http://math.eretrandre.org/tetrationforu...p?tid=1037
http://math.eretrandre.org/tetrationforu...p?tid=1036
In general the number of variables is the dimension.
Reductions can lower the dimension but normally with An integer amount.
For instance the groupring R+(C_4) - iso to the couple (z,r).
Or R+(C_3) iso to the complex Numbers.
This is the closest i could find as An answer to your question for commutative Numbers with Nice properties ...
How this is gonna relate to tetration ...
¯\_(ツ)_/¯
Maybe if you had a plan to relate numbers in Some way ...
Anyway hope this enlightens you.
Regards
Tommy1729
The master
